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Mirrors > Home > MPE Home > Th. List > rankdmr1 | Structured version Visualization version GIF version |
Description: A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankdmr1 | ⊢ (rank‘𝐴) ∈ dom 𝑅1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankidb 9792 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
2 | elfvdm 6926 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → suc (rank‘𝐴) ∈ dom 𝑅1) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) ∈ dom 𝑅1) |
4 | r1funlim 9758 | . . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpri 487 | . . . 4 ⊢ Lim dom 𝑅1 |
6 | limsuc 7835 | . . . 4 ⊢ (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1) |
8 | 3, 7 | sylibr 233 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
9 | rankvaln 9791 | . . 3 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅) | |
10 | limomss 7857 | . . . . 5 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
11 | 5, 10 | ax-mp 5 | . . . 4 ⊢ ω ⊆ dom 𝑅1 |
12 | peano1 7876 | . . . 4 ⊢ ∅ ∈ ω | |
13 | 11, 12 | sselii 3979 | . . 3 ⊢ ∅ ∈ dom 𝑅1 |
14 | 9, 13 | eqeltrdi 2842 | . 2 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
15 | 8, 14 | pm2.61i 182 | 1 ⊢ (rank‘𝐴) ∈ dom 𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2107 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 dom cdm 5676 “ cima 5679 Oncon0 6362 Lim wlim 6363 suc csuc 6364 Fun wfun 6535 ‘cfv 6541 ωcom 7852 𝑅1cr1 9754 rankcrnk 9755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-r1 9756 df-rank 9757 |
This theorem is referenced by: r1rankidb 9796 pwwf 9799 unwf 9802 uniwf 9811 rankr1c 9813 rankelb 9816 rankval3b 9818 rankonid 9821 rankssb 9840 rankr1id 9854 |
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